Concentration and Length Dependence of DNA Looping in Transcriptional Regulation

Suppose that h = 0 and that the underlying graph is bipartite; this means that L can be partitioned into two sets Leven and Lodd such that sites in Leven only have edges to sites in Lodd, and vice versa.

The random geometry of equilibrium phases

The hard-core model analogue of introducing an external ﬁeld in the Ising model on Zd is obtained by replacing the single activity parameter λ by two different activities λeven and λodd, one for sites in Leven and the other for sites in Lodd .

The random geometry of equilibrium phases

We need to assume that the underlying lattice L is bipartite, and thus splits off into two parts, Leven and Lodd .

The random geometry of equilibrium phases

In the statement above, an inﬁnite n-cluster for a conﬁguration σ is an inﬁnite cluster of the subgraph of Zd obtained by keeping only those edges e ∈ B with (σ(x), σ(y)) ∈ Gn, where x is the endpoint of e in the even sublattice Leven and y ∈ Lodd is the other endpoint of e.

The random geometry of equilibrium phases

Taking the P -expectation in (46) we thus obtain on the right-hand side the Bernoulli percolation probability ψp, even (∆ ↔ ∂Λ), where ψp, even is the Bernoulli measure with density p = P (pJ,h x ) on the even sublattice Leven and density 1 on Lodd .

Some alternative procedures to the Fisher test are implemented in R: the Bartlett test (bartlett.test), the Fligner test (fligner.test), the Levene test (levene.test available in the lawstat package), etc.

asympTest: an R package for performing parametric statistical tests and confidence intervals based on the central limit theorem

On a levene type test for equality of two variances.

asympTest: an R package for performing parametric statistical tests and confidence intervals based on the central limit theorem

In 1978 Kabatjanskii and Levenˇstein improved this bound for large d.

Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d>7